Maths - Curl

Curl takes a vector field as an input and gives another vector field as an
output. Curl is a measure of rotation where the which points away from the vortex
according to the right-hand rule. So divergence
tells us how much is going in or out of our infinitesimally small volume, curl
tells how much things are going around the volume.

If the vector field represents velocity, then curl of this will be 2 times
the angular velocity. This
page shows this for a rotating solid object.

We have to be a bit careful about what we mean by rotation here, it is not
just circular movement that will produce a non-zero curl value, imagine we have
a fluid flowing through a pipe in such a way that it is flowing faster in the
centre of the pipe than outside, this will also produce a non-zero curl value.
To visualise this imagine an infinitesimally small paddle wheel in the fluid,
if this would be rotated then curl is non-zero.

Another way to look at it is, curl tells you how much the rate of change of
a function in one direction changes as you move in a different direction.

Curl is defined, in the 3D case (there are relatives to curl for more or less
dimensions), as follows:

Identities

stokes's theorem

Prerequisites

Standards

There are a lot of choices we need to make in mathematics, for example,

Left or right handed coordinate systems.

Vector shown as row or column.

Matrix order.

Direction of x,y and z coordinates.

Euler angle order

Direction of positive angles

Choice of basis for bivectors

Etc. etc.

A lot of these choices are arbitrary as long as we are consistent about it, different authors tend to make different choices and this leads to a lot of confusion. Where standards exist I have tried to follow them (for example x3d and MathML) otherwise I have at least tried to be consistent across the site. I have documented the choices I have made on this page.

Where I can, I have put links to Amazon for commercial software, not
directly related to the software project, but related to the subject being
discussed, click on the appropriate country flag to get more details of
the software or to buy it from them.